LMFDB - Embedded newform 1638.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1638,2,Mod(1,1638)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1638, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1638.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$13.0794958511$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.37228 q^{10} +3.37228 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.37228 q^{17} +7.37228 q^{19} -3.37228 q^{20} -3.37228 q^{22} -9.37228 q^{23} +6.37228 q^{25} +1.00000 q^{26} -1.00000 q^{28} -0.627719 q^{29} +6.74456 q^{31} -1.00000 q^{32} -1.37228 q^{34} +3.37228 q^{35} +1.37228 q^{37} -7.37228 q^{38} +3.37228 q^{40} +2.74456 q^{41} -6.11684 q^{43} +3.37228 q^{44} +9.37228 q^{46} -12.7446 q^{47} +1.00000 q^{49} -6.37228 q^{50} -1.00000 q^{52} +2.74456 q^{53} -11.3723 q^{55} +1.00000 q^{56} +0.627719 q^{58} -2.00000 q^{59} -12.1168 q^{61} -6.74456 q^{62} +1.00000 q^{64} +3.37228 q^{65} -13.4891 q^{67} +1.37228 q^{68} -3.37228 q^{70} -6.74456 q^{71} -2.62772 q^{73} -1.37228 q^{74} +7.37228 q^{76} -3.37228 q^{77} -6.74456 q^{79} -3.37228 q^{80} -2.74456 q^{82} +6.00000 q^{83} -4.62772 q^{85} +6.11684 q^{86} -3.37228 q^{88} +14.7446 q^{89} +1.00000 q^{91} -9.37228 q^{92} +12.7446 q^{94} -24.8614 q^{95} -15.4891 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 9 q^{19} - q^{20} - q^{22} - 13 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} - 7 q^{29} + 2 q^{31} - 2 q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 9 q^{38} + q^{40} - 6 q^{41} + 5 q^{43} + q^{44} + 13 q^{46} - 14 q^{47} + 2 q^{49} - 7 q^{50} - 2 q^{52} - 6 q^{53} - 17 q^{55} + 2 q^{56} + 7 q^{58} - 4 q^{59} - 7 q^{61} - 2 q^{62} + 2 q^{64} + q^{65} - 4 q^{67} - 3 q^{68} - q^{70} - 2 q^{71} - 11 q^{73} + 3 q^{74} + 9 q^{76} - q^{77} - 2 q^{79} - q^{80} + 6 q^{82} + 12 q^{83} - 15 q^{85} - 5 q^{86} - q^{88} + 18 q^{89} + 2 q^{91} - 13 q^{92} + 14 q^{94} - 21 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 + 9 * q^19 - q^20 - q^22 - 13 * q^23 + 7 * q^25 + 2 * q^26 - 2 * q^28 - 7 * q^29 + 2 * q^31 - 2 * q^32 + 3 * q^34 + q^35 - 3 * q^37 - 9 * q^38 + q^40 - 6 * q^41 + 5 * q^43 + q^44 + 13 * q^46 - 14 * q^47 + 2 * q^49 - 7 * q^50 - 2 * q^52 - 6 * q^53 - 17 * q^55 + 2 * q^56 + 7 * q^58 - 4 * q^59 - 7 * q^61 - 2 * q^62 + 2 * q^64 + q^65 - 4 * q^67 - 3 * q^68 - q^70 - 2 * q^71 - 11 * q^73 + 3 * q^74 + 9 * q^76 - q^77 - 2 * q^79 - q^80 + 6 * q^82 + 12 * q^83 - 15 * q^85 - 5 * q^86 - q^88 + 18 * q^89 + 2 * q^91 - 13 * q^92 + 14 * q^94 - 21 * q^95 - 8 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.37228	−1.50813	−0.754065	−	0.656800i	$-0.771910\pi$
$5$	−3.37228	−1.50813	−0.754065	+	0.656800i	$0.771910\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.37228	1.06641
$11$	3.37228	1.01678	0.508391	−	0.861127i	$-0.330241\pi$
$11$	3.37228	1.01678	0.508391	+	0.861127i	$0.330241\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	7.37228	1.69132	0.845659	−	0.533724i	$-0.179208\pi$
$19$	7.37228	1.69132	0.845659	+	0.533724i	$0.179208\pi$
$20$	−3.37228	−0.754065
$21$	0	0
$22$	−3.37228	−0.718973
$23$	−9.37228	−1.95426	−0.977128	−	0.212653i	$-0.931790\pi$
$23$	−9.37228	−1.95426	−0.977128	+	0.212653i	$0.931790\pi$
$24$	0	0
$25$	6.37228	1.27446
$26$	1.00000	0.196116
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−0.627719	−0.116564	−0.0582822	−	0.998300i	$-0.518562\pi$
$29$	−0.627719	−0.116564	−0.0582822	+	0.998300i	$0.518562\pi$
$30$	0	0
$31$	6.74456	1.21136	0.605680	−	0.795709i	$-0.292901\pi$
$31$	6.74456	1.21136	0.605680	+	0.795709i	$0.292901\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−1.37228	−0.235344
$35$	3.37228	0.570020
$36$	0	0
$37$	1.37228	0.225602	0.112801	−	0.993618i	$-0.464018\pi$
$37$	1.37228	0.225602	0.112801	+	0.993618i	$0.464018\pi$
$38$	−7.37228	−1.19594
$39$	0	0
$40$	3.37228	0.533204
$41$	2.74456	0.428629	0.214314	−	0.976765i	$-0.431248\pi$
$41$	2.74456	0.428629	0.214314	+	0.976765i	$0.431248\pi$
$42$	0	0
$43$	−6.11684	−0.932810	−0.466405	−	0.884571i	$-0.654451\pi$
$43$	−6.11684	−0.932810	−0.466405	+	0.884571i	$0.654451\pi$
$44$	3.37228	0.508391
$45$	0	0
$46$	9.37228	1.38187
$47$	−12.7446	−1.85899	−0.929493	−	0.368840i	$-0.879755\pi$
$47$	−12.7446	−1.85899	−0.929493	+	0.368840i	$0.879755\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.37228	−0.901177
$51$	0	0
$52$	−1.00000	−0.138675
$53$	2.74456	0.376995	0.188497	−	0.982074i	$-0.439638\pi$
$53$	2.74456	0.376995	0.188497	+	0.982074i	$0.439638\pi$
$54$	0	0
$55$	−11.3723	−1.53344
$56$	1.00000	0.133631
$57$	0	0
$58$	0.627719	0.0824235
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−12.1168	−1.55140	−0.775701	−	0.631100i	$-0.782604\pi$
$61$	−12.1168	−1.55140	−0.775701	+	0.631100i	$0.782604\pi$
$62$	−6.74456	−0.856560
$63$	0	0
$64$	1.00000	0.125000
$65$	3.37228	0.418280
$66$	0	0
$67$	−13.4891	−1.64796	−0.823979	−	0.566620i	$-0.808251\pi$
$67$	−13.4891	−1.64796	−0.823979	+	0.566620i	$0.808251\pi$
$68$	1.37228	0.166414
$69$	0	0
$70$	−3.37228	−0.403065
$71$	−6.74456	−0.800432	−0.400216	−	0.916421i	$-0.631065\pi$
$71$	−6.74456	−0.800432	−0.400216	+	0.916421i	$0.631065\pi$
$72$	0	0
$73$	−2.62772	−0.307551	−0.153776	−	0.988106i	$-0.549143\pi$
$73$	−2.62772	−0.307551	−0.153776	+	0.988106i	$0.549143\pi$
$74$	−1.37228	−0.159524
$75$	0	0
$76$	7.37228	0.845659
$77$	−3.37228	−0.384307
$78$	0	0
$79$	−6.74456	−0.758823	−0.379411	−	0.925228i	$-0.623873\pi$
$79$	−6.74456	−0.758823	−0.379411	+	0.925228i	$0.623873\pi$
$80$	−3.37228	−0.377033
$81$	0	0
$82$	−2.74456	−0.303086
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−4.62772	−0.501947
$86$	6.11684	0.659596
$87$	0	0
$88$	−3.37228	−0.359486
$89$	14.7446	1.56292	0.781460	−	0.623955i	$-0.214475\pi$
$89$	14.7446	1.56292	0.781460	+	0.623955i	$0.214475\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−9.37228	−0.977128
$93$	0	0
$94$	12.7446	1.31450
$95$	−24.8614	−2.55073
$96$	0	0
$97$	−15.4891	−1.57268	−0.786341	−	0.617792i	$-0.788027\pi$
$97$	−15.4891	−1.57268	−0.786341	+	0.617792i	$0.788027\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	6.37228	0.637228
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	16.8614	1.66140	0.830702	−	0.556718i	$-0.187939\pi$
$103$	16.8614	1.66140	0.830702	+	0.556718i	$0.187939\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−2.74456	−0.266575
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−9.37228	−0.897702	−0.448851	−	0.893607i	$-0.648167\pi$
$109$	−9.37228	−0.897702	−0.448851	+	0.893607i	$0.648167\pi$
$110$	11.3723	1.08430
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−20.0000	−1.88144	−0.940721	−	0.339182i	$-0.889850\pi$
$113$	−20.0000	−1.88144	−0.940721	+	0.339182i	$0.889850\pi$
$114$	0	0
$115$	31.6060	2.94727
$116$	−0.627719	−0.0582822
$117$	0	0
$118$	2.00000	0.184115
$119$	−1.37228	−0.125797
$120$	0	0
$121$	0.372281	0.0338438
$122$	12.1168	1.09701
$123$	0	0
$124$	6.74456	0.605680
$125$	−4.62772	−0.413916
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.37228	−0.295769
$131$	−6.11684	−0.534431	−0.267216	−	0.963637i	$-0.586104\pi$
$131$	−6.11684	−0.534431	−0.267216	+	0.963637i	$0.586104\pi$
$132$	0	0
$133$	−7.37228	−0.639258
$134$	13.4891	1.16528
$135$	0	0
$136$	−1.37228	−0.117672
$137$	16.1168	1.37695	0.688477	−	0.725258i	$-0.258279\pi$
$137$	16.1168	1.37695	0.688477	+	0.725258i	$0.258279\pi$
$138$	0	0
$139$	−10.7446	−0.911342	−0.455671	−	0.890148i	$-0.650601\pi$
$139$	−10.7446	−0.911342	−0.455671	+	0.890148i	$0.650601\pi$
$140$	3.37228	0.285010
$141$	0	0
$142$	6.74456	0.565991
$143$	−3.37228	−0.282004
$144$	0	0
$145$	2.11684	0.175794
$146$	2.62772	0.217472
$147$	0	0
$148$	1.37228	0.112801
$149$	−14.2337	−1.16607	−0.583035	−	0.812447i	$-0.698135\pi$
$149$	−14.2337	−1.16607	−0.583035	+	0.812447i	$0.698135\pi$
$150$	0	0
$151$	1.88316	0.153249	0.0766245	−	0.997060i	$-0.475586\pi$
$151$	1.88316	0.153249	0.0766245	+	0.997060i	$0.475586\pi$
$152$	−7.37228	−0.597971
$153$	0	0
$154$	3.37228	0.271746
$155$	−22.7446	−1.82689
$156$	0	0
$157$	−14.6277	−1.16742	−0.583710	−	0.811963i	$-0.698399\pi$
$157$	−14.6277	−1.16742	−0.583710	+	0.811963i	$0.698399\pi$
$158$	6.74456	0.536569
$159$	0	0
$160$	3.37228	0.266602
$161$	9.37228	0.738639
$162$	0	0
$163$	16.2337	1.27152	0.635760	−	0.771887i	$-0.280687\pi$
$163$	16.2337	1.27152	0.635760	+	0.771887i	$0.280687\pi$
$164$	2.74456	0.214314
$165$	0	0
$166$	−6.00000	−0.465690
$167$	4.11684	0.318571	0.159285	−	0.987233i	$-0.449081\pi$
$167$	4.11684	0.318571	0.159285	+	0.987233i	$0.449081\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.62772	0.354930
$171$	0	0
$172$	−6.11684	−0.466405
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	−6.37228	−0.481699
$176$	3.37228	0.254195
$177$	0	0
$178$	−14.7446	−1.10515
$179$	−8.74456	−0.653599	−0.326800	−	0.945094i	$-0.605970\pi$
$179$	−8.74456	−0.653599	−0.326800	+	0.945094i	$0.605970\pi$
$180$	0	0
$181$	−0.510875	−0.0379730	−0.0189865	−	0.999820i	$-0.506044\pi$
$181$	−0.510875	−0.0379730	−0.0189865	+	0.999820i	$0.506044\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	9.37228	0.690934
$185$	−4.62772	−0.340237
$186$	0	0
$187$	4.62772	0.338412
$188$	−12.7446	−0.929493
$189$	0	0
$190$	24.8614	1.80364
$191$	12.1168	0.876744	0.438372	−	0.898794i	$-0.355555\pi$
$191$	12.1168	0.876744	0.438372	+	0.898794i	$0.355555\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	15.4891	1.11205
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−16.8614	−1.19527	−0.597637	−	0.801767i	$-0.703893\pi$
$199$	−16.8614	−1.19527	−0.597637	+	0.801767i	$0.703893\pi$
$200$	−6.37228	−0.450588
$201$	0	0
$202$	−2.00000	−0.140720
$203$	0.627719	0.0440572
$204$	0	0
$205$	−9.25544	−0.646428
$206$	−16.8614	−1.17479
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	24.8614	1.71970
$210$	0	0
$211$	−23.3723	−1.60901	−0.804507	−	0.593943i	$-0.797570\pi$
$211$	−23.3723	−1.60901	−0.804507	+	0.593943i	$0.797570\pi$
$212$	2.74456	0.188497
$213$	0	0
$214$	2.00000	0.136717
$215$	20.6277	1.40680
$216$	0	0
$217$	−6.74456	−0.457851
$218$	9.37228	0.634771
$219$	0	0
$220$	−11.3723	−0.766719
$221$	−1.37228	−0.0923096
$222$	0	0
$223$	5.48913	0.367579	0.183790	−	0.982966i	$-0.441164\pi$
$223$	5.48913	0.367579	0.183790	+	0.982966i	$0.441164\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	20.0000	1.33038
$227$	11.4891	0.762560	0.381280	−	0.924460i	$-0.375483\pi$
$227$	11.4891	0.762560	0.381280	+	0.924460i	$0.375483\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−31.6060	−2.08404
$231$	0	0
$232$	0.627719	0.0412118
$233$	−21.4891	−1.40780	−0.703900	−	0.710299i	$-0.748560\pi$
$233$	−21.4891	−1.40780	−0.703900	+	0.710299i	$0.748560\pi$
$234$	0	0
$235$	42.9783	2.80359
$236$	−2.00000	−0.130189
$237$	0	0
$238$	1.37228	0.0889518
$239$	−17.4891	−1.13128	−0.565639	−	0.824653i	$-0.691370\pi$
$239$	−17.4891	−1.13128	−0.565639	+	0.824653i	$0.691370\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−0.372281	−0.0239311
$243$	0	0
$244$	−12.1168	−0.775701
$245$	−3.37228	−0.215447
$246$	0	0
$247$	−7.37228	−0.469087
$248$	−6.74456	−0.428280
$249$	0	0
$250$	4.62772	0.292683
$251$	19.3723	1.22277	0.611384	−	0.791334i	$-0.290613\pi$
$251$	19.3723	1.22277	0.611384	+	0.791334i	$0.290613\pi$
$252$	0	0
$253$	−31.6060	−1.98705
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.4891	0.716672	0.358336	−	0.933593i	$-0.383344\pi$
$257$	11.4891	0.716672	0.358336	+	0.933593i	$0.383344\pi$
$258$	0	0
$259$	−1.37228	−0.0852694
$260$	3.37228	0.209140
$261$	0	0
$262$	6.11684	0.377900
$263$	−20.7446	−1.27916	−0.639582	−	0.768723i	$-0.720893\pi$
$263$	−20.7446	−1.27916	−0.639582	+	0.768723i	$0.720893\pi$
$264$	0	0
$265$	−9.25544	−0.568557
$266$	7.37228	0.452024
$267$	0	0
$268$	−13.4891	−0.823979
$269$	−23.4891	−1.43216	−0.716079	−	0.698020i	$-0.754065\pi$
$269$	−23.4891	−1.43216	−0.716079	+	0.698020i	$0.754065\pi$
$270$	0	0
$271$	−4.23369	−0.257178	−0.128589	−	0.991698i	$-0.541045\pi$
$271$	−4.23369	−0.257178	−0.128589	+	0.991698i	$0.541045\pi$
$272$	1.37228	0.0832068
$273$	0	0
$274$	−16.1168	−0.973654
$275$	21.4891	1.29584
$276$	0	0
$277$	15.2554	0.916610	0.458305	−	0.888795i	$-0.348457\pi$
$277$	15.2554	0.916610	0.458305	+	0.888795i	$0.348457\pi$
$278$	10.7446	0.644416
$279$	0	0
$280$	−3.37228	−0.201532
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−30.9783	−1.84147	−0.920733	−	0.390193i	$-0.872408\pi$
$283$	−30.9783	−1.84147	−0.920733	+	0.390193i	$0.872408\pi$
$284$	−6.74456	−0.400216
$285$	0	0
$286$	3.37228	0.199407
$287$	−2.74456	−0.162006
$288$	0	0
$289$	−15.1168	−0.889226
$290$	−2.11684	−0.124305
$291$	0	0
$292$	−2.62772	−0.153776
$293$	17.2554	1.00807	0.504037	−	0.863682i	$-0.331848\pi$
$293$	17.2554	1.00807	0.504037	+	0.863682i	$0.331848\pi$
$294$	0	0
$295$	6.74456	0.392684
$296$	−1.37228	−0.0797622
$297$	0	0
$298$	14.2337	0.824535
$299$	9.37228	0.542013
$300$	0	0
$301$	6.11684	0.352569
$302$	−1.88316	−0.108363
$303$	0	0
$304$	7.37228	0.422829
$305$	40.8614	2.33972
$306$	0	0
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	−3.37228	−0.192154
$309$	0	0
$310$	22.7446	1.29180
$311$	5.48913	0.311260	0.155630	−	0.987815i	$-0.450259\pi$
$311$	5.48913	0.311260	0.155630	+	0.987815i	$0.450259\pi$
$312$	0	0
$313$	−32.9783	−1.86404	−0.932020	−	0.362406i	$-0.881956\pi$
$313$	−32.9783	−1.86404	−0.932020	+	0.362406i	$0.881956\pi$
$314$	14.6277	0.825490
$315$	0	0
$316$	−6.74456	−0.379411
$317$	−16.9783	−0.953594	−0.476797	−	0.879014i	$-0.658202\pi$
$317$	−16.9783	−0.953594	−0.476797	+	0.879014i	$0.658202\pi$
$318$	0	0
$319$	−2.11684	−0.118521
$320$	−3.37228	−0.188516
$321$	0	0
$322$	−9.37228	−0.522297
$323$	10.1168	0.562916
$324$	0	0
$325$	−6.37228	−0.353471
$326$	−16.2337	−0.899101
$327$	0	0
$328$	−2.74456	−0.151543
$329$	12.7446	0.702630
$330$	0	0
$331$	5.25544	0.288865	0.144432	−	0.989515i	$-0.453864\pi$
$331$	5.25544	0.288865	0.144432	+	0.989515i	$0.453864\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−4.11684	−0.225264
$335$	45.4891	2.48534
$336$	0	0
$337$	26.8614	1.46323	0.731617	−	0.681716i	$-0.238766\pi$
$337$	26.8614	1.46323	0.731617	+	0.681716i	$0.238766\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	−4.62772	−0.250973
$341$	22.7446	1.23169
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.11684	0.329798
$345$	0	0
$346$	−10.0000	−0.537603
$347$	31.7228	1.70297	0.851485	−	0.524379i	$-0.175703\pi$
$347$	31.7228	1.70297	0.851485	+	0.524379i	$0.175703\pi$
$348$	0	0
$349$	−11.2554	−0.602490	−0.301245	−	0.953547i	$-0.597402\pi$
$349$	−11.2554	−0.602490	−0.301245	+	0.953547i	$0.597402\pi$
$350$	6.37228	0.340613
$351$	0	0
$352$	−3.37228	−0.179743
$353$	−6.74456	−0.358977	−0.179488	−	0.983760i	$-0.557444\pi$
$353$	−6.74456	−0.358977	−0.179488	+	0.983760i	$0.557444\pi$
$354$	0	0
$355$	22.7446	1.20716
$356$	14.7446	0.781460
$357$	0	0
$358$	8.74456	0.462164
$359$	14.7446	0.778188	0.389094	−	0.921198i	$-0.372788\pi$
$359$	14.7446	0.778188	0.389094	+	0.921198i	$0.372788\pi$
$360$	0	0
$361$	35.3505	1.86055
$362$	0.510875	0.0268510
$363$	0	0
$364$	1.00000	0.0524142
$365$	8.86141	0.463827
$366$	0	0
$367$	25.4891	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$367$	25.4891	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$368$	−9.37228	−0.488564
$369$	0	0
$370$	4.62772	0.240584
$371$	−2.74456	−0.142491
$372$	0	0
$373$	7.48913	0.387772	0.193886	−	0.981024i	$-0.437891\pi$
$373$	7.48913	0.387772	0.193886	+	0.981024i	$0.437891\pi$
$374$	−4.62772	−0.239294
$375$	0	0
$376$	12.7446	0.657251
$377$	0.627719	0.0323292
$378$	0	0
$379$	−17.2554	−0.886352	−0.443176	−	0.896435i	$-0.646148\pi$
$379$	−17.2554	−0.886352	−0.443176	+	0.896435i	$0.646148\pi$
$380$	−24.8614	−1.27536
$381$	0	0
$382$	−12.1168	−0.619952
$383$	33.6060	1.71718	0.858592	−	0.512659i	$-0.171339\pi$
$383$	33.6060	1.71718	0.858592	+	0.512659i	$0.171339\pi$
$384$	0	0
$385$	11.3723	0.579585
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−15.4891	−0.786341
$389$	−14.7446	−0.747579	−0.373790	−	0.927514i	$-0.621942\pi$
$389$	−14.7446	−0.747579	−0.373790	+	0.927514i	$0.621942\pi$
$390$	0	0
$391$	−12.8614	−0.650429
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	6.00000	0.302276
$395$	22.7446	1.14440
$396$	0	0
$397$	16.7446	0.840386	0.420193	−	0.907435i	$-0.361962\pi$
$397$	16.7446	0.840386	0.420193	+	0.907435i	$0.361962\pi$
$398$	16.8614	0.845186
$399$	0	0
$400$	6.37228	0.318614
$401$	12.5109	0.624763	0.312382	−	0.949957i	$-0.398873\pi$
$401$	12.5109	0.624763	0.312382	+	0.949957i	$0.398873\pi$
$402$	0	0
$403$	−6.74456	−0.335971
$404$	2.00000	0.0995037
$405$	0	0
$406$	−0.627719	−0.0311532
$407$	4.62772	0.229387
$408$	0	0
$409$	−2.86141	−0.141487	−0.0707437	−	0.997495i	$-0.522537\pi$
$409$	−2.86141	−0.141487	−0.0707437	+	0.997495i	$0.522537\pi$
$410$	9.25544	0.457093
$411$	0	0
$412$	16.8614	0.830702
$413$	2.00000	0.0984136
$414$	0	0
$415$	−20.2337	−0.993233
$416$	1.00000	0.0490290
$417$	0	0
$418$	−24.8614	−1.21601
$419$	−24.8614	−1.21456	−0.607280	−	0.794488i	$-0.707739\pi$
$419$	−24.8614	−1.21456	−0.607280	+	0.794488i	$0.707739\pi$
$420$	0	0
$421$	15.4891	0.754894	0.377447	−	0.926031i	$-0.376802\pi$
$421$	15.4891	0.754894	0.377447	+	0.926031i	$0.376802\pi$
$422$	23.3723	1.13774
$423$	0	0
$424$	−2.74456	−0.133288
$425$	8.74456	0.424174
$426$	0	0
$427$	12.1168	0.586375
$428$	−2.00000	−0.0966736
$429$	0	0
$430$	−20.6277	−0.994757
$431$	21.2554	1.02384	0.511919	−	0.859034i	$-0.328935\pi$
$431$	21.2554	1.02384	0.511919	+	0.859034i	$0.328935\pi$
$432$	0	0
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	6.74456	0.323749
$435$	0	0
$436$	−9.37228	−0.448851
$437$	−69.0951	−3.30527
$438$	0	0
$439$	15.3723	0.733679	0.366839	−	0.930284i	$-0.380440\pi$
$439$	15.3723	0.733679	0.366839	+	0.930284i	$0.380440\pi$
$440$	11.3723	0.542152
$441$	0	0
$442$	1.37228	0.0652728
$443$	28.9783	1.37680	0.688399	−	0.725332i	$-0.258314\pi$
$443$	28.9783	1.37680	0.688399	+	0.725332i	$0.258314\pi$
$444$	0	0
$445$	−49.7228	−2.35709
$446$	−5.48913	−0.259918
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−5.37228	−0.253534	−0.126767	−	0.991933i	$-0.540460\pi$
$449$	−5.37228	−0.253534	−0.126767	+	0.991933i	$0.540460\pi$
$450$	0	0
$451$	9.25544	0.435822
$452$	−20.0000	−0.940721
$453$	0	0
$454$	−11.4891	−0.539211
$455$	−3.37228	−0.158095
$456$	0	0
$457$	−22.2337	−1.04005	−0.520024	−	0.854152i	$-0.674077\pi$
$457$	−22.2337	−1.04005	−0.520024	+	0.854152i	$0.674077\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	31.6060	1.47364
$461$	−3.60597	−0.167947	−0.0839734	−	0.996468i	$-0.526761\pi$
$461$	−3.60597	−0.167947	−0.0839734	+	0.996468i	$0.526761\pi$
$462$	0	0
$463$	34.3505	1.59640	0.798202	−	0.602389i	$-0.205785\pi$
$463$	34.3505	1.59640	0.798202	+	0.602389i	$0.205785\pi$
$464$	−0.627719	−0.0291411
$465$	0	0
$466$	21.4891	0.995465
$467$	−29.0951	−1.34636	−0.673180	−	0.739478i	$-0.735072\pi$
$467$	−29.0951	−1.34636	−0.673180	+	0.739478i	$0.735072\pi$
$468$	0	0
$469$	13.4891	0.622870
$470$	−42.9783	−1.98244
$471$	0	0
$472$	2.00000	0.0920575
$473$	−20.6277	−0.948464
$474$	0	0
$475$	46.9783	2.15551
$476$	−1.37228	−0.0628984
$477$	0	0
$478$	17.4891	0.799934
$479$	12.3505	0.564310	0.282155	−	0.959369i	$-0.408951\pi$
$479$	12.3505	0.564310	0.282155	+	0.959369i	$0.408951\pi$
$480$	0	0
$481$	−1.37228	−0.0625706
$482$	6.00000	0.273293
$483$	0	0
$484$	0.372281	0.0169219
$485$	52.2337	2.37181
$486$	0	0
$487$	36.4674	1.65249	0.826247	−	0.563308i	$-0.190471\pi$
$487$	36.4674	1.65249	0.826247	+	0.563308i	$0.190471\pi$
$488$	12.1168	0.548504
$489$	0	0
$490$	3.37228	0.152344
$491$	−36.7446	−1.65826	−0.829129	−	0.559057i	$-0.811163\pi$
$491$	−36.7446	−1.65826	−0.829129	+	0.559057i	$0.811163\pi$
$492$	0	0
$493$	−0.861407	−0.0387958
$494$	7.37228	0.331695
$495$	0	0
$496$	6.74456	0.302840
$497$	6.74456	0.302535
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−4.62772	−0.206958
$501$	0	0
$502$	−19.3723	−0.864627
$503$	1.48913	0.0663968	0.0331984	−	0.999449i	$-0.489431\pi$
$503$	1.48913	0.0663968	0.0331984	+	0.999449i	$0.489431\pi$
$504$	0	0
$505$	−6.74456	−0.300129
$506$	31.6060	1.40506
$507$	0	0
$508$	−8.00000	−0.354943
$509$	−37.0951	−1.64421	−0.822106	−	0.569335i	$-0.807201\pi$
$509$	−37.0951	−1.64421	−0.822106	+	0.569335i	$0.807201\pi$
$510$	0	0
$511$	2.62772	0.116243
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−11.4891	−0.506764
$515$	−56.8614	−2.50561
$516$	0	0
$517$	−42.9783	−1.89018
$518$	1.37228	0.0602946
$519$	0	0
$520$	−3.37228	−0.147884
$521$	22.6277	0.991338	0.495669	−	0.868511i	$-0.334923\pi$
$521$	22.6277	0.991338	0.495669	+	0.868511i	$0.334923\pi$
$522$	0	0
$523$	−32.2337	−1.40948	−0.704740	−	0.709465i	$-0.748937\pi$
$523$	−32.2337	−1.40948	−0.704740	+	0.709465i	$0.748937\pi$
$524$	−6.11684	−0.267216
$525$	0	0
$526$	20.7446	0.904506
$527$	9.25544	0.403173
$528$	0	0
$529$	64.8397	2.81912
$530$	9.25544	0.402031
$531$	0	0
$532$	−7.37228	−0.319629
$533$	−2.74456	−0.118880
$534$	0	0
$535$	6.74456	0.291593
$536$	13.4891	0.582641
$537$	0	0
$538$	23.4891	1.01269
$539$	3.37228	0.145254
$540$	0	0
$541$	−28.3505	−1.21888	−0.609442	−	0.792830i	$-0.708607\pi$
$541$	−28.3505	−1.21888	−0.609442	+	0.792830i	$0.708607\pi$
$542$	4.23369	0.181852
$543$	0	0
$544$	−1.37228	−0.0588361
$545$	31.6060	1.35385
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	16.1168	0.688477
$549$	0	0
$550$	−21.4891	−0.916299
$551$	−4.62772	−0.197147
$552$	0	0
$553$	6.74456	0.286808
$554$	−15.2554	−0.648141
$555$	0	0
$556$	−10.7446	−0.455671
$557$	11.4891	0.486810	0.243405	−	0.969925i	$-0.421736\pi$
$557$	11.4891	0.486810	0.243405	+	0.969925i	$0.421736\pi$
$558$	0	0
$559$	6.11684	0.258715
$560$	3.37228	0.142505
$561$	0	0
$562$	10.0000	0.421825
$563$	−6.11684	−0.257794	−0.128897	−	0.991658i	$-0.541144\pi$
$563$	−6.11684	−0.257794	−0.128897	+	0.991658i	$0.541144\pi$
$564$	0	0
$565$	67.4456	2.83746
$566$	30.9783	1.30211
$567$	0	0
$568$	6.74456	0.282996
$569$	−1.25544	−0.0526307	−0.0263153	−	0.999654i	$-0.508377\pi$
$569$	−1.25544	−0.0526307	−0.0263153	+	0.999654i	$0.508377\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−3.37228	−0.141002
$573$	0	0
$574$	2.74456	0.114556
$575$	−59.7228	−2.49061
$576$	0	0
$577$	−23.4891	−0.977865	−0.488933	−	0.872322i	$-0.662614\pi$
$577$	−23.4891	−0.977865	−0.488933	+	0.872322i	$0.662614\pi$
$578$	15.1168	0.628778
$579$	0	0
$580$	2.11684	0.0878972
$581$	−6.00000	−0.248922
$582$	0	0
$583$	9.25544	0.383321
$584$	2.62772	0.108736
$585$	0	0
$586$	−17.2554	−0.712816
$587$	4.51087	0.186184	0.0930919	−	0.995658i	$-0.470325\pi$
$587$	4.51087	0.186184	0.0930919	+	0.995658i	$0.470325\pi$
$588$	0	0
$589$	49.7228	2.04879
$590$	−6.74456	−0.277669
$591$	0	0
$592$	1.37228	0.0564004
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	0	0
$595$	4.62772	0.189718
$596$	−14.2337	−0.583035
$597$	0	0
$598$	−9.37228	−0.383261
$599$	3.88316	0.158661	0.0793307	−	0.996848i	$-0.474722\pi$
$599$	3.88316	0.158661	0.0793307	+	0.996848i	$0.474722\pi$
$600$	0	0
$601$	26.4674	1.07963	0.539813	−	0.841785i	$-0.318495\pi$
$601$	26.4674	1.07963	0.539813	+	0.841785i	$0.318495\pi$
$602$	−6.11684	−0.249304
$603$	0	0
$604$	1.88316	0.0766245
$605$	−1.25544	−0.0510408
$606$	0	0
$607$	1.88316	0.0764349	0.0382175	−	0.999269i	$-0.487832\pi$
$607$	1.88316	0.0764349	0.0382175	+	0.999269i	$0.487832\pi$
$608$	−7.37228	−0.298985
$609$	0	0
$610$	−40.8614	−1.65443
$611$	12.7446	0.515590
$612$	0	0
$613$	−34.8614	−1.40804	−0.704019	−	0.710181i	$-0.748613\pi$
$613$	−34.8614	−1.40804	−0.704019	+	0.710181i	$0.748613\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	3.37228	0.135873
$617$	−49.6060	−1.99706	−0.998531	−	0.0541915i	$-0.982742\pi$
$617$	−49.6060	−1.99706	−0.998531	+	0.0541915i	$0.982742\pi$
$618$	0	0
$619$	−6.11684	−0.245857	−0.122928	−	0.992416i	$-0.539229\pi$
$619$	−6.11684	−0.245857	−0.122928	+	0.992416i	$0.539229\pi$
$620$	−22.7446	−0.913444
$621$	0	0
$622$	−5.48913	−0.220094
$623$	−14.7446	−0.590728
$624$	0	0
$625$	−16.2554	−0.650217
$626$	32.9783	1.31808
$627$	0	0
$628$	−14.6277	−0.583710
$629$	1.88316	0.0750863
$630$	0	0
$631$	−23.6060	−0.939739	−0.469869	−	0.882736i	$-0.655699\pi$
$631$	−23.6060	−0.939739	−0.469869	+	0.882736i	$0.655699\pi$
$632$	6.74456	0.268284
$633$	0	0
$634$	16.9783	0.674292
$635$	26.9783	1.07060
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	2.11684	0.0838067
$639$	0	0
$640$	3.37228	0.133301
$641$	16.2337	0.641192	0.320596	−	0.947216i	$-0.396117\pi$
$641$	16.2337	0.641192	0.320596	+	0.947216i	$0.396117\pi$
$642$	0	0
$643$	−28.8614	−1.13818	−0.569091	−	0.822274i	$-0.692705\pi$
$643$	−28.8614	−1.13818	−0.569091	+	0.822274i	$0.692705\pi$
$644$	9.37228	0.369320
$645$	0	0
$646$	−10.1168	−0.398042
$647$	1.48913	0.0585436	0.0292718	−	0.999571i	$-0.490681\pi$
$647$	1.48913	0.0585436	0.0292718	+	0.999571i	$0.490681\pi$
$648$	0	0
$649$	−6.74456	−0.264747
$650$	6.37228	0.249941
$651$	0	0
$652$	16.2337	0.635760
$653$	−36.8614	−1.44250	−0.721249	−	0.692676i	$-0.756432\pi$
$653$	−36.8614	−1.44250	−0.721249	+	0.692676i	$0.756432\pi$
$654$	0	0
$655$	20.6277	0.805992
$656$	2.74456	0.107157
$657$	0	0
$658$	−12.7446	−0.496835
$659$	−20.7446	−0.808093	−0.404047	−	0.914738i	$-0.632397\pi$
$659$	−20.7446	−0.808093	−0.404047	+	0.914738i	$0.632397\pi$
$660$	0	0
$661$	−0.744563	−0.0289601	−0.0144801	−	0.999895i	$-0.504609\pi$
$661$	−0.744563	−0.0289601	−0.0144801	+	0.999895i	$0.504609\pi$
$662$	−5.25544	−0.204258
$663$	0	0
$664$	−6.00000	−0.232845
$665$	24.8614	0.964084
$666$	0	0
$667$	5.88316	0.227797
$668$	4.11684	0.159285
$669$	0	0
$670$	−45.4891	−1.75740
$671$	−40.8614	−1.57744
$672$	0	0
$673$	−47.0951	−1.81538	−0.907691	−	0.419639i	$-0.862157\pi$
$673$	−47.0951	−1.81538	−0.907691	+	0.419639i	$0.862157\pi$
$674$	−26.8614	−1.03466
$675$	0	0
$676$	1.00000	0.0384615
$677$	−12.5109	−0.480832	−0.240416	−	0.970670i	$-0.577284\pi$
$677$	−12.5109	−0.480832	−0.240416	+	0.970670i	$0.577284\pi$
$678$	0	0
$679$	15.4891	0.594418
$680$	4.62772	0.177465
$681$	0	0
$682$	−22.7446	−0.870934
$683$	6.11684	0.234055	0.117027	−	0.993129i	$-0.462664\pi$
$683$	6.11684	0.234055	0.117027	+	0.993129i	$0.462664\pi$
$684$	0	0
$685$	−54.3505	−2.07663
$686$	1.00000	0.0381802
$687$	0	0
$688$	−6.11684	−0.233202
$689$	−2.74456	−0.104560
$690$	0	0
$691$	22.5109	0.856354	0.428177	−	0.903695i	$-0.359156\pi$
$691$	22.5109	0.856354	0.428177	+	0.903695i	$0.359156\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−31.7228	−1.20418
$695$	36.2337	1.37442
$696$	0	0
$697$	3.76631	0.142659
$698$	11.2554	0.426025
$699$	0	0
$700$	−6.37228	−0.240850
$701$	25.7228	0.971537	0.485769	−	0.874087i	$-0.338540\pi$
$701$	25.7228	0.971537	0.485769	+	0.874087i	$0.338540\pi$
$702$	0	0
$703$	10.1168	0.381564
$704$	3.37228	0.127098
$705$	0	0
$706$	6.74456	0.253835
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	−51.4891	−1.93371	−0.966857	−	0.255317i	$-0.917820\pi$
$709$	−51.4891	−1.93371	−0.966857	+	0.255317i	$0.917820\pi$
$710$	−22.7446	−0.853588
$711$	0	0
$712$	−14.7446	−0.552576
$713$	−63.2119	−2.36731
$714$	0	0
$715$	11.3723	0.425299
$716$	−8.74456	−0.326800
$717$	0	0
$718$	−14.7446	−0.550262
$719$	22.5109	0.839514	0.419757	−	0.907637i	$-0.362115\pi$
$719$	22.5109	0.839514	0.419757	+	0.907637i	$0.362115\pi$
$720$	0	0
$721$	−16.8614	−0.627952
$722$	−35.3505	−1.31561
$723$	0	0
$724$	−0.510875	−0.0189865
$725$	−4.00000	−0.148556
$726$	0	0
$727$	−17.0951	−0.634022	−0.317011	−	0.948422i	$-0.602679\pi$
$727$	−17.0951	−0.634022	−0.317011	+	0.948422i	$0.602679\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	−8.86141	−0.327975
$731$	−8.39403	−0.310464
$732$	0	0
$733$	−19.2554	−0.711216	−0.355608	−	0.934635i	$-0.615726\pi$
$733$	−19.2554	−0.711216	−0.355608	+	0.934635i	$0.615726\pi$
$734$	−25.4891	−0.940821
$735$	0	0
$736$	9.37228	0.345467
$737$	−45.4891	−1.67561
$738$	0	0
$739$	−25.7228	−0.946229	−0.473114	−	0.881001i	$-0.656870\pi$
$739$	−25.7228	−0.946229	−0.473114	+	0.881001i	$0.656870\pi$
$740$	−4.62772	−0.170118
$741$	0	0
$742$	2.74456	0.100756
$743$	24.4674	0.897621	0.448810	−	0.893627i	$-0.351848\pi$
$743$	24.4674	0.897621	0.448810	+	0.893627i	$0.351848\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	−7.48913	−0.274196
$747$	0	0
$748$	4.62772	0.169206
$749$	2.00000	0.0730784
$750$	0	0
$751$	34.9783	1.27637	0.638187	−	0.769881i	$-0.279685\pi$
$751$	34.9783	1.27637	0.638187	+	0.769881i	$0.279685\pi$
$752$	−12.7446	−0.464746
$753$	0	0
$754$	−0.627719	−0.0228602
$755$	−6.35053	−0.231120
$756$	0	0
$757$	37.2119	1.35249	0.676245	−	0.736676i	$-0.263606\pi$
$757$	37.2119	1.35249	0.676245	+	0.736676i	$0.263606\pi$
$758$	17.2554	0.626746
$759$	0	0
$760$	24.8614	0.901818
$761$	1.02175	0.0370384	0.0185192	−	0.999829i	$-0.494105\pi$
$761$	1.02175	0.0370384	0.0185192	+	0.999829i	$0.494105\pi$
$762$	0	0
$763$	9.37228	0.339299
$764$	12.1168	0.438372
$765$	0	0
$766$	−33.6060	−1.21423
$767$	2.00000	0.0722158
$768$	0	0
$769$	−25.3723	−0.914948	−0.457474	−	0.889223i	$-0.651246\pi$
$769$	−25.3723	−0.914948	−0.457474	+	0.889223i	$0.651246\pi$
$770$	−11.3723	−0.409829
$771$	0	0
$772$	2.00000	0.0719816
$773$	−29.0951	−1.04648	−0.523239	−	0.852186i	$-0.675276\pi$
$773$	−29.0951	−1.04648	−0.523239	+	0.852186i	$0.675276\pi$
$774$	0	0
$775$	42.9783	1.54382
$776$	15.4891	0.556027
$777$	0	0
$778$	14.7446	0.528618
$779$	20.2337	0.724947
$780$	0	0
$781$	−22.7446	−0.813864
$782$	12.8614	0.459923
$783$	0	0
$784$	1.00000	0.0357143
$785$	49.3288	1.76062
$786$	0	0
$787$	9.88316	0.352296	0.176148	−	0.984364i	$-0.443636\pi$
$787$	9.88316	0.352296	0.176148	+	0.984364i	$0.443636\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	−22.7446	−0.809215
$791$	20.0000	0.711118
$792$	0	0
$793$	12.1168	0.430282
$794$	−16.7446	−0.594242
$795$	0	0
$796$	−16.8614	−0.597637
$797$	7.72281	0.273556	0.136778	−	0.990602i	$-0.456325\pi$
$797$	7.72281	0.273556	0.136778	+	0.990602i	$0.456325\pi$
$798$	0	0
$799$	−17.4891	−0.618721
$800$	−6.37228	−0.225294
$801$	0	0
$802$	−12.5109	−0.441774
$803$	−8.86141	−0.312712
$804$	0	0
$805$	−31.6060	−1.11396
$806$	6.74456	0.237567
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	−21.2554	−0.747301	−0.373651	−	0.927569i	$-0.621894\pi$
$809$	−21.2554	−0.747301	−0.373651	+	0.927569i	$0.621894\pi$
$810$	0	0
$811$	−14.1168	−0.495709	−0.247855	−	0.968797i	$-0.579726\pi$
$811$	−14.1168	−0.495709	−0.247855	+	0.968797i	$0.579726\pi$
$812$	0.627719	0.0220286
$813$	0	0
$814$	−4.62772	−0.162201
$815$	−54.7446	−1.91762
$816$	0	0
$817$	−45.0951	−1.57768
$818$	2.86141	0.100047
$819$	0	0
$820$	−9.25544	−0.323214
$821$	−3.72281	−0.129927	−0.0649635	−	0.997888i	$-0.520693\pi$
$821$	−3.72281	−0.129927	−0.0649635	+	0.997888i	$0.520693\pi$
$822$	0	0
$823$	25.7228	0.896641	0.448320	−	0.893873i	$-0.352022\pi$
$823$	25.7228	0.896641	0.448320	+	0.893873i	$0.352022\pi$
$824$	−16.8614	−0.587395
$825$	0	0
$826$	−2.00000	−0.0695889
$827$	−28.6277	−0.995483	−0.497742	−	0.867325i	$-0.665837\pi$
$827$	−28.6277	−0.995483	−0.497742	+	0.867325i	$0.665837\pi$
$828$	0	0
$829$	−18.6277	−0.646967	−0.323484	−	0.946234i	$-0.604854\pi$
$829$	−18.6277	−0.646967	−0.323484	+	0.946234i	$0.604854\pi$
$830$	20.2337	0.702322
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	1.37228	0.0475467
$834$	0	0
$835$	−13.8832	−0.480446
$836$	24.8614	0.859850
$837$	0	0
$838$	24.8614	0.858823
$839$	12.7446	0.439991	0.219996	−	0.975501i	$-0.429396\pi$
$839$	12.7446	0.439991	0.219996	+	0.975501i	$0.429396\pi$
$840$	0	0
$841$	−28.6060	−0.986413
$842$	−15.4891	−0.533791
$843$	0	0
$844$	−23.3723	−0.804507
$845$	−3.37228	−0.116010
$846$	0	0
$847$	−0.372281	−0.0127917
$848$	2.74456	0.0942487
$849$	0	0
$850$	−8.74456	−0.299936
$851$	−12.8614	−0.440883
$852$	0	0
$853$	41.2119	1.41107	0.705535	−	0.708675i	$-0.250707\pi$
$853$	41.2119	1.41107	0.705535	+	0.708675i	$0.250707\pi$
$854$	−12.1168	−0.414630
$855$	0	0
$856$	2.00000	0.0683586
$857$	−44.9783	−1.53643	−0.768214	−	0.640193i	$-0.778854\pi$
$857$	−44.9783	−1.53643	−0.768214	+	0.640193i	$0.778854\pi$
$858$	0	0
$859$	10.7446	0.366600	0.183300	−	0.983057i	$-0.441322\pi$
$859$	10.7446	0.366600	0.183300	+	0.983057i	$0.441322\pi$
$860$	20.6277	0.703399
$861$	0	0
$862$	−21.2554	−0.723963
$863$	−26.9783	−0.918350	−0.459175	−	0.888346i	$-0.651855\pi$
$863$	−26.9783	−0.918350	−0.459175	+	0.888346i	$0.651855\pi$
$864$	0	0
$865$	−33.7228	−1.14661
$866$	−10.0000	−0.339814
$867$	0	0
$868$	−6.74456	−0.228925
$869$	−22.7446	−0.771556
$870$	0	0
$871$	13.4891	0.457062
$872$	9.37228	0.317385
$873$	0	0
$874$	69.0951	2.33718
$875$	4.62772	0.156445
$876$	0	0
$877$	−7.02175	−0.237108	−0.118554	−	0.992948i	$-0.537826\pi$
$877$	−7.02175	−0.237108	−0.118554	+	0.992948i	$0.537826\pi$
$878$	−15.3723	−0.518789
$879$	0	0
$880$	−11.3723	−0.383360
$881$	42.8614	1.44404	0.722019	−	0.691873i	$-0.243214\pi$
$881$	42.8614	1.44404	0.722019	+	0.691873i	$0.243214\pi$
$882$	0	0
$883$	−24.6277	−0.828789	−0.414394	−	0.910097i	$-0.636007\pi$
$883$	−24.6277	−0.828789	−0.414394	+	0.910097i	$0.636007\pi$
$884$	−1.37228	−0.0461548
$885$	0	0
$886$	−28.9783	−0.973543
$887$	27.2119	0.913687	0.456844	−	0.889547i	$-0.348980\pi$
$887$	27.2119	0.913687	0.456844	+	0.889547i	$0.348980\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	49.7228	1.66671
$891$	0	0
$892$	5.48913	0.183790
$893$	−93.9565	−3.14413
$894$	0	0
$895$	29.4891	0.985713
$896$	1.00000	0.0334077
$897$	0	0
$898$	5.37228	0.179275
$899$	−4.23369	−0.141201
$900$	0	0
$901$	3.76631	0.125474
$902$	−9.25544	−0.308172
$903$	0	0
$904$	20.0000	0.665190
$905$	1.72281	0.0572682
$906$	0	0
$907$	33.9565	1.12751	0.563754	−	0.825943i	$-0.309357\pi$
$907$	33.9565	1.12751	0.563754	+	0.825943i	$0.309357\pi$
$908$	11.4891	0.381280
$909$	0	0
$910$	3.37228	0.111790
$911$	−40.1168	−1.32913	−0.664565	−	0.747230i	$-0.731383\pi$
$911$	−40.1168	−1.32913	−0.664565	+	0.747230i	$0.731383\pi$
$912$	0	0
$913$	20.2337	0.669637
$914$	22.2337	0.735425
$915$	0	0
$916$	6.00000	0.198246
$917$	6.11684	0.201996
$918$	0	0
$919$	31.2119	1.02959	0.514793	−	0.857314i	$-0.327869\pi$
$919$	31.2119	1.02959	0.514793	+	0.857314i	$0.327869\pi$
$920$	−31.6060	−1.04202
$921$	0	0
$922$	3.60597	0.118756
$923$	6.74456	0.222000
$924$	0	0
$925$	8.74456	0.287519
$926$	−34.3505	−1.12883
$927$	0	0
$928$	0.627719	0.0206059
$929$	−26.7446	−0.877461	−0.438730	−	0.898619i	$-0.644572\pi$
$929$	−26.7446	−0.877461	−0.438730	+	0.898619i	$0.644572\pi$
$930$	0	0
$931$	7.37228	0.241617
$932$	−21.4891	−0.703900
$933$	0	0
$934$	29.0951	0.952021
$935$	−15.6060	−0.510370
$936$	0	0
$937$	47.4891	1.55140	0.775701	−	0.631101i	$-0.217396\pi$
$937$	47.4891	1.55140	0.775701	+	0.631101i	$0.217396\pi$
$938$	−13.4891	−0.440436
$939$	0	0
$940$	42.9783	1.40180
$941$	−13.2554	−0.432115	−0.216057	−	0.976381i	$-0.569320\pi$
$941$	−13.2554	−0.432115	−0.216057	+	0.976381i	$0.569320\pi$
$942$	0	0
$943$	−25.7228	−0.837650
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	20.6277	0.670665
$947$	36.8614	1.19783	0.598917	−	0.800811i	$-0.295598\pi$
$947$	36.8614	1.19783	0.598917	+	0.800811i	$0.295598\pi$
$948$	0	0
$949$	2.62772	0.0852994
$950$	−46.9783	−1.52418
$951$	0	0
$952$	1.37228	0.0444759
$953$	22.5109	0.729199	0.364599	−	0.931164i	$-0.381206\pi$
$953$	22.5109	0.729199	0.364599	+	0.931164i	$0.381206\pi$
$954$	0	0
$955$	−40.8614	−1.32224
$956$	−17.4891	−0.565639
$957$	0	0
$958$	−12.3505	−0.399028
$959$	−16.1168	−0.520440
$960$	0	0
$961$	14.4891	0.467391
$962$	1.37228	0.0442441
$963$	0	0
$964$	−6.00000	−0.193247
$965$	−6.74456	−0.217115
$966$	0	0
$967$	42.1168	1.35439	0.677193	−	0.735805i	$-0.263196\pi$
$967$	42.1168	1.35439	0.677193	+	0.735805i	$0.263196\pi$
$968$	−0.372281	−0.0119656
$969$	0	0
$970$	−52.2337	−1.67712
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	10.7446	0.344455
$974$	−36.4674	−1.16849
$975$	0	0
$976$	−12.1168	−0.387851
$977$	7.88316	0.252205	0.126102	−	0.992017i	$-0.459753\pi$
$977$	7.88316	0.252205	0.126102	+	0.992017i	$0.459753\pi$
$978$	0	0
$979$	49.7228	1.58915
$980$	−3.37228	−0.107724
$981$	0	0
$982$	36.7446	1.17257
$983$	16.3505	0.521501	0.260750	−	0.965406i	$-0.416030\pi$
$983$	16.3505	0.521501	0.260750	+	0.965406i	$0.416030\pi$
$984$	0	0
$985$	20.2337	0.644699
$986$	0.861407	0.0274328
$987$	0	0
$988$	−7.37228	−0.234544
$989$	57.3288	1.82295
$990$	0	0
$991$	−20.2337	−0.642744	−0.321372	−	0.946953i	$-0.604144\pi$
$991$	−20.2337	−0.642744	−0.321372	+	0.946953i	$0.604144\pi$
$992$	−6.74456	−0.214140
$993$	0	0
$994$	−6.74456	−0.213925
$995$	56.8614	1.80263
$996$	0	0
$997$	43.4891	1.37731	0.688657	−	0.725087i	$-0.258201\pi$
$997$	43.4891	1.37731	0.688657	+	0.725087i	$0.258201\pi$
$998$	20.0000	0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.a.v.1.1	✓	2
3.2	odd	2		1638.2.a.x.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1638.2.a.v.1.1	✓	2	1.1	even	1	trivial
1638.2.a.x.1.2	yes	2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.37228 q^{10} +3.37228 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.37228 q^{17} +7.37228 q^{19} -3.37228 q^{20} -3.37228 q^{22} -9.37228 q^{23} +6.37228 q^{25} +1.00000 q^{26} -1.00000 q^{28} -0.627719 q^{29} +6.74456 q^{31} -1.00000 q^{32} -1.37228 q^{34} +3.37228 q^{35} +1.37228 q^{37} -7.37228 q^{38} +3.37228 q^{40} +2.74456 q^{41} -6.11684 q^{43} +3.37228 q^{44} +9.37228 q^{46} -12.7446 q^{47} +1.00000 q^{49} -6.37228 q^{50} -1.00000 q^{52} +2.74456 q^{53} -11.3723 q^{55} +1.00000 q^{56} +0.627719 q^{58} -2.00000 q^{59} -12.1168 q^{61} -6.74456 q^{62} +1.00000 q^{64} +3.37228 q^{65} -13.4891 q^{67} +1.37228 q^{68} -3.37228 q^{70} -6.74456 q^{71} -2.62772 q^{73} -1.37228 q^{74} +7.37228 q^{76} -3.37228 q^{77} -6.74456 q^{79} -3.37228 q^{80} -2.74456 q^{82} +6.00000 q^{83} -4.62772 q^{85} +6.11684 q^{86} -3.37228 q^{88} +14.7446 q^{89} +1.00000 q^{91} -9.37228 q^{92} +12.7446 q^{94} -24.8614 q^{95} -15.4891 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - q^{5} - 2 q^{7} - 2 q^{8} + q^{10} + q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 9 q^{19} - q^{20} - q^{22} - 13 q^{23} + 7 q^{25} + 2 q^{26} - 2 q^{28} - 7 q^{29} + 2 q^{31} - 2 q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 9 q^{38} + q^{40} - 6 q^{41} + 5 q^{43} + q^{44} + 13 q^{46} - 14 q^{47} + 2 q^{49} - 7 q^{50} - 2 q^{52} - 6 q^{53} - 17 q^{55} + 2 q^{56} + 7 q^{58} - 4 q^{59} - 7 q^{61} - 2 q^{62} + 2 q^{64} + q^{65} - 4 q^{67} - 3 q^{68} - q^{70} - 2 q^{71} - 11 q^{73} + 3 q^{74} + 9 q^{76} - q^{77} - 2 q^{79} - q^{80} + 6 q^{82} + 12 q^{83} - 15 q^{85} - 5 q^{86} - q^{88} + 18 q^{89} + 2 q^{91} - 13 q^{92} + 14 q^{94} - 21 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - q^5 - 2 * q^7 - 2 * q^8 + q^10 + q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 + 9 * q^19 - q^20 - q^22 - 13 * q^23 + 7 * q^25 + 2 * q^26 - 2 * q^28 - 7 * q^29 + 2 * q^31 - 2 * q^32 + 3 * q^34 + q^35 - 3 * q^37 - 9 * q^38 + q^40 - 6 * q^41 + 5 * q^43 + q^44 + 13 * q^46 - 14 * q^47 + 2 * q^49 - 7 * q^50 - 2 * q^52 - 6 * q^53 - 17 * q^55 + 2 * q^56 + 7 * q^58 - 4 * q^59 - 7 * q^61 - 2 * q^62 + 2 * q^64 + q^65 - 4 * q^67 - 3 * q^68 - q^70 - 2 * q^71 - 11 * q^73 + 3 * q^74 + 9 * q^76 - q^77 - 2 * q^79 - q^80 + 6 * q^82 + 12 * q^83 - 15 * q^85 - 5 * q^86 - q^88 + 18 * q^89 + 2 * q^91 - 13 * q^92 + 14 * q^94 - 21 * q^95 - 8 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more